Lower Bounds on Nonnegative Signed Domination Parameters in Graphs

Let \(1 \leq k \leq n\) be a positive integer. A {\em nonnegative signed \(k\)-subdominating function} is a function \(f:V(G) \rightarrow \{-1,1\}\) satisfying \(\sum_{u\in N_G[v]}f(u) \geq 0\) for at least \(k\) vertices \(v\) of \(G\). The value \(\min\sum_{v\in V(G)} f(v)\), taking over all nonnegative signed \(k\)-subdominating functions \(f\) of \(G\), is called the {\em nonnegative signed \(k\)-subdomination number} of \(G\) and denoted by \(\gamma^{NN}_{ks}(G)\). When \(k=|V(G)|\), \(\gamma^{NN}_{ks}(G)=\gamma^{NN}_s(G)\) is the {\em nonnegative signed domination number}, introduced in \cite{HLFZ}. In this paper, we investigate several sharp lower bounds of \(\gamma^{NN}_s(G)\), which extend some presented lower bounds on \(\gamma^{NN}_s(G)\). We also initiate the study of the nonnegative signed \(k\)-subdomination number in graphs and establish some sharp lower bounds for \(\gamma^{NN}_{ks}(G)\) in terms of order and the degree sequence of a graph \(G\).